Analysis

[Kels]

I am trying to work through an exam review before Friday. I have figured out every problem except for one. It's driving me crazy. Was hoping someone could help me to figure it out...?

What does it mean for a function to be continuous on a set?

My choices:

1. A function is continuous on a set S if for every [math]\epsilon\nobreak>0[/math]\nobreak, there is [math]\delta\nobreak>0[/math]\nobreak such that |f(x)-f)c)|<[math]\epsilon[/math]\nobreak whenever x, c [math]\in[/math]\nobreakS and |x-c|<[math]\delta[/math]\nobreak

2. A function [math]f:S\to\R[/math] is continuous on the set S if for every sequence {[math]x_n[/math]} in S, the sequence {[math]f(x_n)[/math]} converges

3. A function [math]f: S\to\R[/math] is continuous at the point c[math]\in[/math]S if for every [math]\epsilon[/math]>0, there is [math]\delta>0[/math] such that |f(x)-f(c)|<[math]\epsilon[/math] whenever x [math]\in[/math] S ad |x-c|<[math]\delta[/math] if f continuous at all c [math]\in[/math] S then f is continuous on set S

4. A function [math]f:S\to\R[/math] is continuous on the set S if the left limit of f is equal to the right limit of f at every point of S

5. A function [math]f:S\to\R[/math] is continuous on the set S if f is continuous at every cluster point of S

I am leaning towards #3

Any help is appreciated

 

[Zermelo]

Well, you should tell us what is your previous knowledge. What is the definition of continuity you learned in your class? What epsilon delta definitions did you learn?
You’re right about 3), but you should understand why it’s true. Can you write an epsilon delta definition for limits, and any definition of continuity?